348 Chapter 8 Confidence Intervals never exactly equal to the true values they are estimating. They are almost always off — sometimes by a little, sometimes by a lot. It is unlikely that the mean level of PM for all of 2007–2008 is exactly equal to 20.21; it is somewhat more or less than that. In order for a point estimate to be useful, it is necessary to describe just how close to the true value it is likely to be. To do this, statisticians construct confidence intervals. A confidence interval gives a range of values that is likely to contain the true value being estimated. To construct a confidence interval, we put a plus-or-minus number on the point estimate. So, for example, we might estimate that the population mean is 20.21 ± 2.0, or equivalently, that the population mean is between 18.21 and 22.21. The interval 20.21±2.0, or equivalently, (18.21, 22.21), is a confidence interval for the population mean. One of the benefits of confidence intervals is that they come with a measure of the level of confidence we can have that they actually cover the true value being estimated. For example, we will show that we can be 95% confident that the population mean PM level during the winter of 2007–2008 is in the interval 20.21 ± 2.27, or equivalently, between 17.94 and 22.48. If we want more confidence, we can widen the interval. For example, we will learn how to show that we can be 99% confident that the population mean PM level is in the interval 20.21 ± 3.11, or equivalently, between 17.10 and 23.32. In the case study at the end of the chapter, we will use confidence intervals to further study the effects of the Libby stove replacement program. There are many different situations in which confidence intervals can be constructed. The correct method to use varies from situation to situation. In this chapter, we will describe the methods that are appropriate in several of the most commonly encountered situations. SECTION 8.1 Confidence Intervals for a Population Mean, Standard Deviation Known Objectives 1. Construct and interpret confidence intervals for a population mean when the population standard deviation is known 2. Find critical values for confidence intervals 3. Describe the relationship between the confidence level and the margin of error 4. Find the sample size necessary to obtain a confidence interval of a given width 5. Distinguish between confidence and probability Objective 1 Construct and interpret confidence intervals for a population mean when the population standard deviation is known Estimating a Population Mean How can we measure the reading ability of elementary school students? The No Child Left Behind Act, signed into law in 2002, requires schools to regularly assess the proficiency of students in subjects such as reading and math. In a certain school district, administrators are trying out a new experimental approach to teach reading to fourth-graders. A simple random sample of 100 fourth-graders is selected to take part in the program. At the end of the program, the students are given a standardized reading test. On the basis of past results, it is known that scores on this test have a population standard deviation of �� = 15. Recall: A parameter is a numerical summary of a population, such as a population mean �� or a population proportion p. The sample mean score for the 100 students was ̄x = 67.30. The administrators want to estimate what the mean score �� would be if the entire population of fourth-graders in the district had enrolled in the program. The best estimate for the population mean is the sample mean, ̄x = 67.30. The sample mean is a point estimate, because it is a single number. DEFINITION A point estimate is a single number that is used to estimate the value of an unknown parameter. It is very unlikely that the point estimate ̄x is exactly equal to the population mean ��. Therefore, in order for the estimate to be useful, we must describe how close it is likely to be. For example, if we think that ̄x = 67.30 is likely to be within 1 point of the population mean, we would estimate �� with the interval 66.30 < �� < 68.30. This could also be written
navidi_monk_elementary_statistics_2e_ch7-9
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